At this point in time specifically, a $1000 laptop (the comparable general purpose option) is going to be a mid-to-upper entry-level one. Because it’s supposed to be general purpose, it won’t have hardware chosen to maximize gaming potential. Likely all of these options would only allow a RAM or hard drive upgrade, as well. This puts them upon somewhat equal footing with the Steam Machine, except I would hazard a guess that the Steam Machine would still do better at gaming due to the hardware selection they had available at production time.
And if you mean building a $1000 gaming Linux box, good luck not going over that limit with current RAM, GPU, and SSD prices.

Right now, you can’t build a comparable desktop for $1000, unless you go for one of those BC250 builds. RAM, SSDs, and GPUs would push the starting price (from zero, mind you) to around $1500. Even just the 2x16 “budget RAM” I bought four years ago is close to $250-$300 for the set now.
On the other hand, you can pretty easily find a comparable entry level laptop that can do gaming for $1000. Will it beat a $1000 desktop machine from a few years ago? Will it beat a true gaming laptop? No, but that’s not what we’re comparing.